Question

Consider a sample with a mean of 30 and a standard deviation of 6. Use Chebyshev's theorem to determine the percentage of the data within each of the following ranges(to the nearest whole number).

a) 10 to 50, at least ____%

b)15 to 45, at least ___%

c) 22 to 38, at least ___%

d) 17 to 43, at least ___%

e) 14 to 46, at least ___%

Answer #1

**Chebyshev's theorem:-**

SD = 6

M= 30

**CASE 1 ( 10 TO 50 )**

K= UL - M/SD = 50 - 30/6 = 3.33

1- 1/k^{2} = 1-1/3.33^{2}= 1-1/11.0889 = 11.0889
-1/11.0889 = 10.0889/11.0889 = 0.90 =90 %

**CASE 2 (15 TO 45)**

K = UL -M /SD = 45 - 30 / 6 = 2.5

1-1/K^{2} = 1- 1/ 2.5^{2} = 1-1/6.25 = 6.25 - 1
/ 6.25 = 5.25 /6.25 = 0.84 = 84%

**CASE 3 (22 TO 38 )**

K= UL - M / SD = 38 - 30/6 = 1.33

1-1/K^{2} = 1 - 1/ 1.33^{2}= 1- 1/ 1.7778
=0.4375=43.75%

**CASE 4 (17 TO 43)**

K = UL -M / SD = 43 - 30 / 6 = 2.16667

1 - 1/ K^{2} = 1- 1/ 2.16667^{2} =
0.786983=78.70%

**CASE 5 ( 14 TO 46)**

K= UL - M / SD = 46 - 30 / 6 = 2.66667

1 - 1 / K^{2} = 1 - 1 / 2.66667^{2} =
0.8594=85.94%

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40
40
42
39
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40
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